John, Smith and Kate start at same time, same point and in same direction to run around a circular ground. John completes a round in 250 seconds, Smith in 300 seconds and Kate in 150 seconds. Find after what time will they meet again at the starting point?

To find out when John, Smith, and Kate will meet again at the starting point, we need to determine the least common multiple (LCM) of the times it takes each of them to complete one round. - John completes a round in 250 seconds. - Smith completes a round in 300 seconds. - Kate completes a round in 150 seconds. Now, we will find the LCM of 250, 300, and 150. 1. **Prime factorization**: - \( 250 = 2 \times 5^3 \) - \( 300 = 2^2 \times 3 \times 5^2 \) - \( 150 = 2 \times 3 \times 5^2 \) 2. **Identify the highest power of each prime**: - For \( 2 \): the highest power is \( 2^2 \) (from 300). - For \( 3 \): the highest power is \( 3^1 \) (from both 300 and 150). - For \( 5 \): the highest power is \( 5^3 \) (from 250). 3. **Calculate the LCM**: \[ \text{LCM} = 2^2 \times 3^1 \times 5^3 \] \[ = 4 \times 3 \times 125 \] \[ = 12 \times 125 \] \[ = 1500 \] Thus, John, Smith, and Kate will all meet again at the starting point after **1500 seconds**.